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Journal of Geometric Mechanics

2013 , Volume 5 , Issue 3

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A setting for higher order differential equation fields and higher order Lagrange and Finsler spaces
Ioan Bucataru
2013, 5(3): 257-279 doi: 10.3934/jgm.2013.5.257 +[Abstract](353) +[PDF](482.3KB)
Abstract:
We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order $2k$ in terms of a semi-basic $1$-form of order $k$. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [15] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projective metrizability in terms of homogeneous semi-basic $1$-forms. Such a semi-basic $1$-form is the Poincaré-Cartan $1$-form of a higher order Finsler function, while the potential of such semi-basic $1$-form is a higher order Finsler function.
Smooth perfectness for the group of diffeomorphisms
Stefan Haller, Tomasz Rybicki and Josef Teichmann
2013, 5(3): 281-294 doi: 10.3934/jgm.2013.5.281 +[Abstract](389) +[PDF](410.1KB)
Abstract:
Given a result of Herman, we provide a new elementary proof of the fact that the connected component of the identity in the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism $g$, which is sufficiently close to the identity, can be represented as a product of four commutators, $g=[h_1,k_1]\circ\cdots\circ[h_4,k_4]$, where the factors $h_i$ can be chosen to depend smoothly on $g$, and $k_i=\exp(X_i)$ are flows at time one of complete vector fields $X_i$ which are independent of $g$.
Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation
Weipeng Hu, Zichen Deng and Yuyue Qin
2013, 5(3): 295-318 doi: 10.3934/jgm.2013.5.295 +[Abstract](493) +[PDF](9277.5KB)
Abstract:
The soliton interactions, especially the soliton resonance phenomena of the (2+1)-dimensional Boussinesq equation have been investigated numerically in this paper. Based on the Bridges's multi-symplectic idea, the multi-symplectic formulations with several conservation laws for the (2+1)-dimensional Boussinesq equation are presented firstly. Then, a forty-five points implicit multi-symplectic scheme is constructed. Finally, according to the soliton resonance condition, numerical experiments on the two-soliton solution of the (2+1)-dimensional Boussinesq equation for simulating the soliton interaction phenomena, especially the soliton resonance are reported. From the results of the numerical experiments, it can be concluded that the multi-symplectic scheme can simulate the soliton resonance phenomena perfectly, which can be used to make further investigation on the interaction and the energy distribution of gravity waves, and evaluate the impact on the ship traffic on the surface of water.
On Euler's equation and 'EPDiff'
David Mumford and Peter W. Michor
2013, 5(3): 319-344 doi: 10.3934/jgm.2013.5.319 +[Abstract](671) +[PDF](661.6KB)
Abstract:
We study a family of approximations to Euler's equation depending on two parameters $\epsilon,η \ge 0$. When $\epsilon = η = 0$ we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group ${Diff}_{H^\infty}(\mathbb{R}^n)$ or, if $\epsilon = 0$, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by $$ ||v||_{\epsilon,η} = \int_{\mathbb{R}^n} \langle L_{\epsilon,η} v, v \rangle\, dx $$ where $L_{\epsilon,η} = (I-\frac{η^2}{p} \triangle)^p \circ (I-\frac {1}{\epsilon^2} \nabla \circ div)$. All geodesic equations are locally well-posed, and the $L_{\epsilon,η}$-equation admits solutions for all time if $η > 0$ and $p\ge (n+3)/2$. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to ``vortex-solitons", also called ``landmarks" in imaging science, and to new numeric approximations to fluids.
The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups
Anahita Eslami Rad and Enrique G. Reyes
2013, 5(3): 345-364 doi: 10.3934/jgm.2013.5.345 +[Abstract](377) +[PDF](473.3KB)
Abstract:
We present explicit formal solutions to the systems of equations in two independent variables $t_m$, $x$, $m =1,2,\dots$, of the Kadomtsev-Petviashvili hierarchy. The main tools used are a Birkhoff-like factorization of formal Lie groups due to M. Mulase, and the classical theory of A.G. Reyman and M.A. Semenov-Tian-Shansky on the integration of Hamiltonian systems on coadjoint orbits using $r$-matrices. Our paper also contains full proofs of Mulase's results.
Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms
Yuri B. Suris
2013, 5(3): 365-379 doi: 10.3934/jgm.2013.5.365 +[Abstract](447) +[PDF](403.3KB)
Abstract:
Recently, Lobb and Nijhoff initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multi-dimensional consistency. In the present work, we follow this line of research and develop a Lagrangian theory of integrable one-dimensional systems. We give a complete solution of the following problem: one looks for a function of several variables (interpreted as multi-time) which delivers critical points to the action functionals obtained by integrating a Lagrangian 1-form along any smooth curve in the multi-time. The Lagrangian 1-form is supposed to depend on the first jet of the sought-after function. We derive the corresponding multi-time Euler-Lagrange equations and show that, under the multi-time Legendre transform, they are equivalent to a system of commuting Hamiltonian flows. Involutivity of the Hamilton functions turns out to be equivalent to closeness of the Lagrangian 1-form on solutions of the multi-time Euler-Lagrange equations. In the discrete time context, the analogous extremal property turns out to be characteristic for systems of commuting symplectic maps. For one-parameter families of commuting symplectic maps (Bäcklund transformations), we show that their spectrality property, introduced by Kuznetsov and Sklyanin, is equivalent to the property of the Lagrangian 1-form to be closed on solutions of the multi-time Euler-Lagrange equations, and propose a procedure of constructing Lax representations with the only input being the maps themselves.

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